$(20$ points$)$ Virtual Reality Goggle Inventory. Galaxy Co$.$ sells virtual reality $($VR$)$ goggles targeted to customers who like to play video games. Galaxy procures each pair of goggles for $$150$ from its supplier and sells each pair of goggles for $$300.$ Monthly demand for the VR goggles is a normal random variable with a mean of $160$ units and a standard deviation of $40$ units. At the beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to $140$ goggles. If the monthly demand is less than $140,$ Galaxy pays $$20$ per pair of goggles that remains in inventory at the end of the month. If the monthly demand exceeds $140,$ Galaxy sells only the $140$ pairs of goggles in stock. Galaxy assigns a shortage cost of $$40$ for each unit of demand that is unsatisfied to represent a loss$-$of$-$goodwill among its customers.

a$.$ Construct a simulation model to compute the monthly profit and unmet demand for Galaxy Co$.$

b$.$ What is the average monthly profit resulting from its policy of stocking $140$ pairs of goggles at the beginning of each month?

c$.$ What is the proportion of months in which demand is completely satisfied?

d$.$ Compare the average monthly profit resulting from stocking $105,140,$ and $175$ pairs of goggles at the beginning of each month. Which amount results in the largest average monthly profit?